Kepler's first law
Gravitational Fields - OCR A-Level Physics
- Kepler's first lawAll planets orbit the Sun in ellipses with the Sun at one focus.: Planets move in elliptical orbits with the Sun at one focus.
- Kepler's second lawA line joining a planet and the Sun sweeps out equal areas in equal times.: A line joining a planet to the Sun sweeps out equal areas in equal times. (Planets move faster when closer to the Sun.)
- Kepler's third lawThe square of the orbital period is proportional to the cube of the semi-major axis: T² ∝ r³.: \(T^{2}\) \propto \(r^{3}\), or equivalently \(T^{2}\)/\(r^{3}\) is constant for all objects orbiting the same central mass.
$$T^2 = \frac{4\pi^2}{GM}r^3$$
- Derived by equating gravitational force to centripetal forceThe resultant force directed towards the centre of a circular path that causes an object to move in a circle. It is not a separate force but the net force providing circular motion.: GMm/\(r^{2}\) = m($2\pi/$T)^2 r.
- Rearranging: \(T^{2}\) = ($4\pi$^2/GM)\(r^{3}\).
- This can be used to find the mass of the central body if T and r are known: M = $4\pi$^2 \(r^{3}\)/(GT^2).
- For circular orbits, the orbital speed is v = $\sqrt{GM/r}.$ Higher orbits have slower speeds.
Examiner Tips and Tricks
- To find the mass of a planet from satellite data: measure T and r, then use M = $4\pi$^2 \(r^{3}\) / (GT^2).
- This is how we determine the masses of planets and stars.